To model the height of the heart on the bike tire as it travels in a circular path, we can use a sine or cosine function. In this case, let's consider the heart's vertical position in relation to time.
Explanation of Choosing Sine or Cosine:
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Form of the Function:
- A sine function (y = A * sin(Bx + C) + D) typically starts at the midline (D) and oscillates above and below this line. A cosine function (y = A * cos(Bx + C) + D) starts at the maximum value (A + D).
- Because the heart starts at the maximum height when the bike tire starts (the topmost point of the wheel), a cosine function is appropriate because it starts at the peak of the cycle.
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Period of the Function:
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A full revolution of the bike wheel corresponds to one complete cycle of the cosine function. To find the period, we need to determine how far the wheel travels in one minute:
- Circumference of the Wheel: \[ \text{Circumference} = \pi \times \text{Diameter} = \pi \times 24 \text{ inches} = 24\pi \text{ inches} \]
- Distance in One Minute (with 70 revolutions): \[ \text{Distance in One Minute} = 70 \times 24\pi \text{ inches} = 1680\pi \text{ inches} \]
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Conversion to Feet: As there are 12 inches in a foot, \[ 1680\pi \text{ inches} = \frac{1680\pi}{12} \text{ feet} = 140\pi \text{ feet} \]
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Height of the Wheel:
- The radius of the wheel is half of the diameter: \[ r = \frac{24}{2} = 12 \text{ inches} \]
- As the wheel rolls, the vertical height will oscillate between:
- Maximum height = radius = 12 inches (at the top of the tire)
- Minimum height = 0 inches (when in contact with the ground)
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Vertical Shift and Amplitude:
- The midline (D) of the function is at the radius level (12 inches) while the amplitude (A) is also 12 inches since the height varies by 12 inches above and below the midline.
Equation of the Function:
Now we can write the equation. The standard form for the cosine function, with the information we have:
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Amplitude (A) = 12
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Period (P) = circumference = \( 24\pi \) inches
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Since the bike makes 70 revolutions per minute, we want to express this in terms of time (in minutes). The input (x) will be in terms of time in minutes.
The value of B can be calculated using: \[ B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{24\pi} = \frac{1}{12} \]
The resulting function that represents the height \( h(t) \) of the heart above the ground over time \( t \) (in minutes) is:
\[ h(t) = 12 \cos\left(\frac{1}{12}t\right) + 12 \]
Summary:
- Function Style: Cosine
- Final Equation: \( h(t) = 12 \cos\left(\frac{1}{12}t\right) + 12 \)
- This equation shows the oscillating behavior of the heart's height on the tire as it rides along its path.
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